Midterm1 - Mathematics Department UCLA T Richthammer spring...

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Mathematics Department, UCLA T. Richthammer spring 09, midterm 1 Apr 27, 2009 Midterm 1: Math 170B Probability, Sec. 1 1. Let Z = X + Y , where X, Y are independent RVs with uniform distribution on [0 , 1]. (a) Calculate E ( Z ), V ( Z ) and the PDF of Z . (b) State for which of the results in (a) you really need the independence of X, Y . Answer: (a) E ( X ) = R 1 0 xdx = 1 2 and E ( X 2 ) = R 1 0 x 2 dx = 1 3 , i.e. V ( X ) = 1 3 - 1 4 = 1 12 . So E ( Z ) = E ( X ) + E ( Y ) = 1 and V ( Z ) = V ( X ) + V ( Y ) = 1 6 . Using the formula from the lecture the PDF f of Z is f ( z ) = R f X ( x ) f Y ( z - x ) dx = R 1 { 0 x 1 , 0 z - x 1 } dx = R 1 { max { 0 ,z - 1 }≤ x min { 1 ,z }} dx , which is = R z 0 dx = z for 0 z 1 and = R 1 z - 1 dx = 2 - z for 1 z 2, so f Z ( z ) = z 1 { 0 z 1 } + (2 - z )1 { 1 <z 2 } . Without using the formula from the lecture: For 0 c 2 we have F Z ( c ) = P ( Z c ) = λ 2 ( A c ), where A c is the subset of a square below the line y = c - x . For c 1 we have λ 2 ( A c ) = 1 2 c 2 , and for c 1 we have λ 2 ( A c ) = 1 - 1 2 (2 - c ) 2 . So f Z ( c ) = F ± Z ( c ) = c for c 1 and = 2 - c for c > 1, and we get the same answer as above. (b) Independence is needed for the variance and the PDF, but not for the expectation.
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This note was uploaded on 07/07/2010 for the course MATH 170B 170B taught by Professor Richthammer during the Winter '10 term at UCLA.

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Midterm1 - Mathematics Department UCLA T Richthammer spring...

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